On the H-triangle of generalised nonnesting partitions

Transcription

1 FPSAC 204, Chicago, USA DMTCS proc. AT, 204, On the H-triangle of generalised nonnesting partitions Marko Thiel Department of Mathematics, University of Vienna, Austria Abstract. With a crystallographic root system Φ, there are associated two Catalan objects, the set of nonnesting partitions NN(Φ), and the cluster complex (Φ). These possess a number of enumerative coincidences, many of which are captured in a surprising identity, first conjectured by Chapoton. We prove this conjecture, and indicate its generalisation for the Fuß-Catalan objects NN (k) (Φ) and (k) (Φ), conjectured by Armstrong. Résumé. À un système de racines cristallographique, on associe deux objets de Catalan: l ensemble des partitions non-emboîtées NN(Φ), et le complexe d amas (Φ). Ils possèdent de nombreuses coïncidences énumératives, plusieurs d entre elles étant capturées dans une identité surprenante, conjecturée par Chapoton. Nous démontrons cette conjecture, et indiquons sa généralisation pour les objets de Fuß-Catalan NN (k) (Φ) et (k) (Φ), conjecturée par Armstrong. Keywords: nonnesting partitions, noncrossing partitions, cluster complex, Coxeter-Catalan objects Introduction For a crystallographic root system Φ, there are three well-known Coxeter-Catalan objects [Arm09]: the set of noncrossing partitions N C(Φ), the set of nonnesting partitions N N(Φ) and the cluster complex (Φ). The former two and the set of facets of the latter are all counted by the same numbers, the Coxeter-Catalan numbers Cat(Φ). For the root system of type A n, these reduce to objects counted by the classical Catalan numbers C n = n+( 2n ) n, namely the set of noncrossing partitions of [n] = {, 2,..., n}, the set of nonnesting partitions of [n] and the set of triangulations of a convex (n + 2)-gon, respectively. Each of these Coxeter-Catalan objects has a generalisation [Arm09], a Fuß-Catalan object defined for each positive integer k. These are the set of k-divisible noncrossing partitions NC (k) (Φ), the set of k-generalised nonnesting partitions NN (k) (Φ) and the generalised cluster complex (k) (Φ). They specialise to the corresponding Coxeter-Catalan objects when k =. The former two and the set of facets of the latter are counted by Fuß-Catalan numbers Cat (k) (Φ), which specialise to the classical Fuß-Catalan numbers C (k) n = kn+ ( (k+)n n ) in type An. marko.thiel@univie.ac.at c 204 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

2 84 Marko Thiel The enumerative coincidences do not end here. Chapoton defined the M-triangle, the H-triangle and the F -triangle, which are polynomials in two variables that encode refined enumerative information on NC(Φ), NN(Φ) and (Φ) respectively [Cha04, Cha06]. This allowed him to formulate the M = F conjecture [Cha04, Conjecture ] and the H = F conjecture [Cha06, Conjecture 6.] relating these polynomials through invertible transformations of variables. These conjectures were later generalised to the corresponding Fuß-Catalan objects by Armstrong [Arm09, Conjecture ]. The M = F conjecture was first proven by Athanasiadis [Ath07] for k =, and later by Krattenthaler [Kra06a, Kra06b] and Tzanaki [Tza08] for k >. In this extended abstract, we prove the H = F conjecture in the k = case. The proof of the generalised conjecture of Armstrong can be found in the full version [Thi4]. 2 Definitions and the Main Result Let Φ = Φ(S) be a crystallographic root system with a simple system S. Then Φ = Φ + Φ + is the disjoint union of the set of positive roots Φ + and the set of negative roots Φ +. Every positive root can be written uniquely as a linear combination of the simple roots and all coefficients of this linear combination are nonnegative integers. For further background on root systems, see [Hum90]. Define the root order on Φ + by β α if and only if β α S N, that is, β α if and only if β α can be written as a linear combination of simple roots with nonnegative integer coefficients. The set of positive roots Φ + with this partial order is called the root poset. A set of pairwise incomparable elements of the root poset is called an antichain. The support of a root β Φ + is the set of all α S with α β. We define the set of nonnesting partitions NN(Φ) of Φ as the set of antichains in the root poset of Φ. Let us define the H-triangle [Cha06, Section 6] as H Φ (x, y) = x A y A S. A NN(Φ) Let Φ = Φ + S be the set of almost positive roots of Φ. Then there exists a symmetric binary relation called compatibility [FZ03, Definition 3.4] on Φ such that all negative simple roots are pairwise compatible and for α S and β Φ +, α is compatible with β if and only if α is not in the support of β. Define a simplicial complex (Φ) as the set of all subsets A Φ such that all almost positive roots in A are pairwise compatible. This is the cluster complex of Φ. This simplicial complex is pure, all facets have cardinality n, where n = S is the rank of Φ. Let us define the F -triangle [Cha04, Section 2] as F Φ (x, y) = F (Φ) x F Φ+ y F S = l,m f l,m (Φ)x l y m. Consider the Weyl group W = W (Φ) of the root system Φ. A standard Coxeter element in W is a

3 On the H-triangle of generalised nonnesting partitions 85 product of all the simple reflections of W in some order. A Coxeter element is any element of W that is conjugate to a standard Coxeter element. Let T denote the set of reflections in W. For w W, define the absolute length l T (w) of w as the minimal l such that w = t t 2 t l for some t, t 2,..., t l T. Define the absolute order on W by u T v if and only if l T (u) + l T (u v) = l T (v). Fix a Coxeter element c W. We define the set of noncrossing partitions NC(Φ) of Φ as the interval [e, c] in the absolute order. We drop the choice of the Coxeter element c from the notation, since a different choice of Coxeter element results in a different but isomorphic poset. Let us define the M-triangle [Cha04, Section 3] as M Φ (x, y) = u,v NC(Φ) µ(u, v)x rk(u) y rk(v), where rk is the rank function of the graded poset NC(Φ) and µ is its Möbius function. As mentioned in the introduction, NC(Φ), NN(Φ) and the set of facets of (Φ) are all counted by the same number Cat(Φ). But more is true: define the Narayana number Nar(Φ, i) as the number of elements of NC(Φ) of rank i [Arm09, Definition 3.5.4]. The number of antichains in the root poset of cardinality i also equals Nar(Φ, i) [Ath05, Proposition 5., Remark 5.2]. Let (h 0, h,..., h n ) be the h-vector of (Φ), defined by the relation n i=0 h i x n i = l,m f l,m (x ) n (l+m). Then h n i = Nar(Φ, i) for all i {0,,..., n} [FR05, Theorem 3.2]. The main result of this extended abstract is the following theorem, conjectured by Chapoton. Theorem If Φ is a crystallographic root system of rank n, then H Φ (x, y) = (x ) n + (y )x F Φ,. x x In order to prove Theorem, we first find a combinatorial bijection for nonnesting partitions that leads to a differential equation for the H-triangle analogous to one known for the F -triangle. Using both of these differential equations and induction on the rank n, we prove Theorem by showing that the derivatives with respect to y of both sides of the equation as well as their specialisations at y = agree. After proving Theorem, we use it together with the M = F (ex-)conjecture to relate the H-triangle to the M-triangle. 3 Proof of the Main Result To prove Theorem, we show that the derivatives with respect to y of both sides of the equation agree, as well as their specialisations at y =. To do this, we need the following lemmas.

4 86 Marko Thiel Lemma If Φ is a crystallographic root system of rank n, then H Φ (x, ) = (x ) n F Φ x,. x Proof: We have (x ) n F Φ x, = n f l,m (x ) n (l+m) = h i x n i, x l,m i=0 where (h 0, h,..., h n ) is the h-vector of (Φ). So [x i ](x ) n F Φ x, = h n i = Nar(Φ, i), x by [FR05, Theorem 3.2]. But [x i ]H Φ (x, ) = Nar(Φ, i), by [Ath05, Proposition 5., Remark 5.2]. Lemma 2 ([Cha04, Proposition 3]) If Φ is a crystallographic root system of rank n, then y F Φ(S)(x, y) = F Φ(S\{α}) (x, y). α S Lemma 3 For every simple root α S, there exists a bijection Θ from the set of nonnesting partitions A NN(Φ(S)) with α A to NN(Φ(S\{α})), such that Θ(A) = A and Θ(A) S\{α} = A S for all A NN(Φ(S)). Proof: Define Θ(A) = A\{α}. If β A and β α, then β and α are incomparable, since A is an antichain. So α is not in the support of β. So β Φ(S\{α}). Clearly Θ(A) = A\{α} is an antichain in the root poset of Φ(S\{α}), so Θ is well defined. It is also clear that Θ(A) = A and Θ(A) S\{α} = A S for all A NN(Φ(S)). Define the map Ψ : NN(Φ(S\{α})) NN(Φ(S)) by Ψ(A) = A {α}. If A NN(Φ(S\{α})) and β A, then α is not in the support of β, so α and β are incomparable. Thus Ψ(A) = A {α} is an antichain in the root poset of Φ(S), so Ψ is well-defined. Clearly Ψ is the inverse of Θ, so Θ is a bijection. Lemma 4 If Φ is a crystallographic root system of rank n, then y H Φ(S)(x, y) = x H Φ(S\{α}) (x, y). α S

5 On the H-triangle of generalised nonnesting partitions 87 Proof: Say h l,m (Φ) = [x l y m ]H Φ (x, y). We wish to show that mh l,m (Φ(S)) = α S h l,m (Φ(S\{α})). So we seek a bijection Θ from the set of pairs (A, α) with A NN(Φ(S)) and α A S to the set of pairs (α, A ) with α S and A NN(Φ(S\{α })) such that if Θ(A, α) = (A, α ), then A = A and A S\{α } = A S. Such a bijection is given in Lemma 3. We are now in a position to prove Theorem. Proof of Theorem : We proceed by induction on n. If n = 0, both sides are equal to, so the result holds. If n > 0, y H Φ(S)(x, y) = x H Φ(S\{α}) (x, y), α S by Lemma 4. By induction hypothesis, this is further equal to x (x ) n + (y )x F Φ(S\{α}),, x x α S which equals by Lemma 2. But by Lemma, so y (x + (y )x )n F Φ(S), x x H Φ (x, ) = (x ) n F Φ x, x H Φ (x, y) = (x ) n + (y )x F Φ,, x x since the derivatives with respect to y as well as the specialisations at y = of both sides agree. 4 Consequences and generalisations We may also recover the original statement of the conjecture, due to Chapoton. Corollary ([Cha06, Conjecture 6.]) If Φ is a crystallographic root system of rank n, then ( ) x H Φ (x, y) = ( x) n F Φ x, xy. x Proof: We have [Cha04, Proposition 5] F Φ (x, y) = ( ) n F Φ ( x, y). () Substitute () into Theorem to get the result. Using the M = F (ex-)conjecture, we can also relate the H-triangle to the M-triangle.

6 88 Marko Thiel Corollary 2 If Φ is a crystallographic root system of rank n, then ( y H Φ (x, y) = ( + (y )x) n M Φ y, (y )x + (y )x Proof: We have [Kra06a, Conjecture FM] [Ath07, Theorem.] ( + y F Φ (x, y) = y n M Φ y x, y x ). (2) y Substitute (2) into Theorem to get the result. Lemma, Lemma 2, Lemma 4, Theorem and Corollary 2 all generalise to the corresponding Fuß- Catalan objects NN (k) (Φ), NC (k) (Φ) and (k) (Φ). For proofs of these more general results and further consequences, see the full version of the article [Thi4]. References [Arm09] Drew Armstrong. Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups. Memoirs of the American Mathematical Society, 202, [Ath05] Christos A. Athanasiadis. On a Refinement of the Generalized Catalan Numbers for Weyl Groups. Transactions of the American Mathematical Society, 357:79 96, ). [Ath07] Christos A. Athanasiadis. On some enumerative aspects of generalized associahedra. European Journal of Combinatorics, 28:208 25, [Cha04] Frédéric Chapoton. Enumerative Properties of Generalized Associahedra. Séminaire Lotharingien de Combinatiore, 5, [Cha06] Frédéric Chapoton. Sur le nombre de réflexions pleines dans les groupes de coxeter finis. Bulletin of the Belgian Mathematical Society, 3: , [FR05] [FZ03] Sergey Fomin and Nathan Reading. Generalized Cluster Complexes and Coxeter Combinatorics. International Mathematics Research Notices, 44: , Sergey Fomin and Andrei Zelevinsky. Y -Systems and Generalized Associahedra. Annals of Mathematics, 58:977 08, [Hum90] James E. Humphreys. Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge, 990. [Kra06a] Christian Krattenthaler. The F -triangle of the generalised cluster complex. Topics in Discrete Mathematics, pages 93 26, [Kra06b] Christian Krattenthaler. The M-triangle of generalised non-crossing partitions for the types E 7 and E 8. Séminaire Lotharingien de Combinatoire, 54, 2006.

7 On the H-triangle of generalised nonnesting partitions 89 [Thi4] [Tza08] Marko Thiel. On the H-triangle of generalised nonnesting partitions. European Journal of Combinatorics, 39: , 204. Eleni Tzanaki. Faces of Generalized Cluster Complexes and Noncrossing Partitions. SIAM Journal on Discrete Mathematics, 22:5 30, 2008.

8 90 Marko Thiel